Physica B 194-196 (1994) 523-524 North-Holland PflYSICA A Self-Bound Waveflmction for Clusters of 4He

Shiwei Zhang, ~ M.H. Kalos, ~ G.V. Chester, ~ S.A.Vitiello, b and L. Reatto b

~Department of Physics, ( 'ornefl University, Ithaca, New York 14853, USA

bDipart imeuto di Fisica, Universitg degli Studi Milano 20 133 Milano, I taly

Using the ideas of shadow wavefunctions, we propose a new form for the ground state of droplets of 4He. The shadow-shadow interaction is modified to reflect the variation of local density in an inhomogeneous system• We (:an thereby get a bound state without one-body form factors. When the cluster is large, the bulk wavefunction is restored. The binding energy of the system is in satisfactory agreement with previous computations.

1. I N T R O D U C T I O N

We have studied the ground state of 4He droplets using the variational Monte Carlo (VMC) method with a new and improved wave- function based on tile shadow wavefunction pro- posed by Vitietlo el al. [1] for bulk 4He. Tile system we consider has N 4tte a toms interacting by the two-body potential of Aziz et al. [2].

The shadow wavefunction can be written as

~',(R) = { E ( R , S)dS, (1) I

where =(R, S) is given by

exp [ - E Ur ( r i j ) - E c(rk - s k )2_ Z Us(Sij )](2) i< j k i<j

Here R = { r l , r2 , . . . , rN} and rij is the dis- taI'lce between a toms i and j , and similarly for the shadow variables. The two-body terms Ur and u.~ are variational functions and the coupling constant c is a variational parame- ter. The energy can be easily evaluated by Monte Carlo (MC) methods as the average of HE(R, S2)/E(R, S2) if points {R, S1,S2} are sampled from --(R, St)-'a(R, $2). We note that Us does not appear in the estimator. This results in great flexibility in the wavefunctiou, since we can buihl into the shadow correlation function a structure that is difficult to deal with analytically and yet both simple and powerful when included in the MC. sampling.

In the existing VMC, work [3], the trial wave- functions for liquid 4He droplets contain a prod-

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uct of one-body factors for every particle multi- plied by a typical bulk liquid wavefunction. While one-body form factors may work well in produc- ing a good estimate of the ground state energy, they are somewhat arbi trary and lack physical motivation other than the a priori knowledge that the system forms a bound state. We propose a new droplet wavefunction which does not contain one-body form factors. This wavefunction gives more freedom to the surface profile of the droplet.

2. W A V E F U N C T I O N F O R M

We use a new shadow-shadow pseudopotential:

~ls(Si j ) b u l k / ~ r = u s Lsij)+(f(ni)+f(nj))~vt~it(slj).(3)

The first term on the right-hand side is the pseu- dopotential for the bulk phase. The function Vt~il(sij) is essentially 0 at small distances and asymptotically linear with the slope a parame- ter. The coefficients multiplying this tail te rm are functions of the local densities ni and nj of shadow variables i and j , respectively• The lo- cal density ni can simply be measured by the number of shadow particles that are within a cer- tain distance r,~ (a parameter) of shadow parti- cle i. Qualitatively, the variational function f (n) should decrease from 1 to 0 in a shape similar to the Fermi function, as the local density increases from 0 to the maximum corresponding to the cen- ter of the drop.

It is reasonable to expect the long-range be- havior of the wavefunction in this inhomogeneous

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,vslem to depend on the local environment. More ~peofically, in the interior of the droplet, particles ,~xperience very weak attraction from any partic- ular direction and "'feel" little net elfect. On the other hand, for ~t particle at the edge of the drop the weak attractions from other particles add up to give a net contribution that pulls it toward the center of the drop. The above wavefunction reflects this, where the local density n measures tim location of the particle and f modifies the long-range behavior of the two-body correlation accordingly.

3. R E S U L T S A N D D I S C U S S I O N

We have applied this wavefunction to droplets with N = 20 and N = 108 particles. The bulk shadow pseudopotential was chosen to be a rescaled Aziz form [4]. Ttle pseudopotential for tile real particles used the optimized basis set re- sult of Vitiello and Schmidt [5] for bulk 4He.

The local density construction can be handled straightforwardly in the MC sampling process. Variational MC estimators for expectation val- ues remain the same as that for the bulk shadow wavefunction. Some care has to be taken in choosing the parameter rn, which must be large enough to overcome the strong fluctuation in par- t icle positions.

Table 1 shows some preliminary results on the variational energy, together with exact results from Green's function Monte Carlo calculations [6] as well as previous VMC results [7] using wave- functions with one-body form factors and two and three-body correlations. We see that they are in very good agreement. That these excellent energy values were obtained without a very exhaustive parameter search is a good measure of the qual- ity of the wavefunctions, although energy values are not tile main concern of this work.

The method we have described applies our knowledge of these systems to improve systemati- cally the few-body correlations. In fact only two- body correlations are used explicitly. As an ad- ditional desirable feature, the wavefunction auto- matically interpolates between the bulk form and a droplet form as the system size changes. The idea of introducing correlations which depend on

Table 1 Calculated variational energies (in K) compared with previous VMC and exact results.

i i ii i u ii i immn

N present previous VMC z exact ~ i

20 - 1 6 0 6 ( 1 5 ) -1 .573(1)

108 -a .426(30) -3 .470(5)

- 1.627(3)

-3 .57{1) i . , !

the local density should be easily generalizable to other inhomogeneous systems.

4. A C K N O W L E D G E M E N T S

This work is supported by the Condensed Mat- ter Theory Program of the NSF under Grant DMR-9200469 and by Consorzio INFM. The Cot- nell Theory Center is funded by the U.S. National Science Foundation, by New York State, by IBM, and by Cornell University.

REFERENCES

i. S.A. Vitiello, K.J. Runge, and M.H. Kalos, Phys. Rev. Lett. 60, 1970, (1988); S.A. Vi- tiello, K.J. Runge, G.V. Chester, and M.H. Kalos, Phys. Rev. B 42, 228. (1990).

2. R.A. Aziz, V.P.S. Nain. J.S. Carley, W.L. Taylor, and G.T. McConville, J. Chem. Phys. 70, 4330, (1979).

3. See, e.g., K.E. Schmidt and D.M. Ceperley, in The Monte Carlo Method in Condensed Mat- ter Physics, ed. by K. Binder (Springer Verlag 1992), and references therein.

4. T. MacFarland, S.A. Vitiello, and L. Reatto, J. Low Temp. Phys. 89. 433, (1992).

5. S.A. Vitiello and K.E. Schmidt, Phys. Rev. B 46, 5442, (1992).

6. V.R. Pandharipande, J.G. Zabolitzky, S.C. Pieper, R.B. Wiringa, and U. Helmbrecht, Phys. Rev. Lett. 50, 1676, (1983).

7. V.R. Pandharipande, S.C. Pieper, and R.B. Wiringa, Phys. Rev. B 34, 4571, (1986).